0cm10mm on the central configurations of the n body

On the central configurationsof the N–body problem

JAUME LLIBRE

Universitat Autònoma de Barcelona

Segona Jornada de Sistemes Dinàmics a CatalunyaBarcelona, Octubre 4, 2017

JAUME LLIBRE Universitat Autònoma de Barcelona On the central configurations of the N–body problem

The contens

1 General introduction to central configurations

2 Central configurations of the coorbital satellite problem

3 Central configurations of p nested n–gons

4 Central configurations of p nested regular polyhedra

5 Another open problem on central configurations

The contens

The N–body problem

The N–body problem consists in study the motion of N pointlikemasses, interacting among themselves through no other forcesthan their mutual gravitational attraction according to Newton’sgravitational law.

Equations of motion

The equations of motion are

mk r′′k =n∑

j=1,j 6=i

(rj − rk ),

for k = 1, . . . ,N. Here,

G is the gravitational constant,

rk ∈ R3 is the position vector of the punctual mass mk in aninertial system,

rjk is the euclidean distance between mj and mk .

Equations of motion

mk r′′k =n∑

j=1,j 6=i

(rj − rk ),

for k = 1, . . . ,N. Here,

Equations of motion

mk r′′k =n∑

j=1,j 6=i

(rj − rk ),

for k = 1, . . . ,N. Here,

Equations of motion

mk r′′k =n∑

j=1,j 6=i

(rj − rk ),

for k = 1, . . . ,N. Here,

Inertial barycentric system

The center of mass of the system satisfies

N∑k=1

m1 + . . .+ mN= at + b,

where a and b are constant vectors.

We can consider that the center of mass is at the origin of theinertial system; i.e.

N∑k=1

mk rk = 0.

A such inertial system is called inertial barycentric system. Inthe rest of the talk we will work in an inertial barycentric system.

N∑k=1

m1 + . . .+ mN= at + b,

N∑k=1

mk rk = 0.

N∑k=1

m1 + . . .+ mN= at + b,

N∑k=1

mk rk = 0.

Homographic solutions

Since the general solution of the N–body problem cannot begiven, great importance has been attached from the verybeginning to the search for particular solutions where the Nmass points fulfilled certain initial conditions.

A homographic solution of the N–body problem is a solutionsuch that the configuration formed by the N–bodies at theinstant t (with respect to an inertial barycentric system) remainssimilar to itself as t varies.

Two configurations are similar if we can pass from one to theother doing a dilatation and/or a rotation.

Homographic solutions of the 3–body problem

The first three homographic solutions where found in 1767 byEuler in the 3–body problem. For these three solutions theconfiguration of the 3 bodies is collinear.

In 1772 Lagrange found two additional homographic solutionsin the 3–body problem. Now, the configuration formed by the 3bodies is an equilateral triangle.

Homographic solutions of the 3–body problem

The first three homographic solutions where found in 1767 byEuler in the 3–body problem. For these three solutions theconfiguration of the 3 bodies is collinear.

In 1772 Lagrange found two additional homographic solutionsin the 3–body problem. Now, the configuration formed by the 3bodies is an equilateral triangle.

Central configurations

At a given instant t = t0 the configuration of the N–bodies iscentral if the gravitational acceleration r′′k acting on every masspoint mk is proportional with the same constant ofproportionality to its position rk (referred to an inertialbarycentric system); i.e.

r′′k =n∑

j=1,j 6=k

(rj − rk ) = λrk , for k = 1, . . . ,N.

Laplace Theorem: The configuration of the N bodies in ahomographic solution is central at any instant of time.

It is important to note that homographic solutions with rotationand eventually with a dilatation only exist for planar centralconfigurations.

For spatial central configurations all the homographic solutionsonly have a dilation.

r′′k =n∑

j=1,j 6=k

(rj − rk ) = λrk , for k = 1, . . . ,N.

r′′k =n∑

j=1,j 6=k

(rj − rk ) = λrk , for k = 1, . . . ,N.

r′′k =n∑

j=1,j 6=k

(rj − rk ) = λrk , for k = 1, . . . ,N.

Where appear the central configurations (I)

Central configurations of the N–body problem are importantbecause:

(1) They allow to compute all the homographic solutions.

(2) If the N bodies are going to a simultaneous collision, thenthe particles tend to a central configuration.

(3) If the N bodies are going simultaneously at infinity inparabolic motion (i.e. the radial velocity of each particle tendsto zero as the particle tends to infinity), then the particles tendto a central configuration.

Where appear the central configurations (II)

(4) There is a relation between central configurations and thebifurcations of the hypersurfaces of constant energy andangular momentum.

(5) Central configurations provides good places for theobservation in the solar system, for instance, SOHO project.

(6) . . .

Classes of central configurations

If we have a central configuration, a dilatation and a rotation(centered at the center of masses) of it, provide another centralconfiguration.

We say that two central configurations are related if we canpass from one to another through a dilation and a rotation. Thisrelation is an equivalence.

In what follows we will talk about the classes of centralconfigurations defined by this equivalent relation.

Collinear central configurations

In 1910 Moulton characterized the number of collinear centralconfigurations by showing that there exist exactly n!/2 classesof collinear central configurations of the n–body problem for agiven set of positive masses, one for each possible ordering ofthe particles.

F.R. MOULTON, The straight line solutions of n bodies, Ann. ofMath. 12 (1910), 1–17.

Collinear central configurations

In 1910 Moulton characterized the number of collinear centralconfigurations by showing that there exist exactly n!/2 classesof collinear central configurations of the n–body problem for agiven set of positive masses, one for each possible ordering ofthe particles.

F.R. MOULTON, The straight line solutions of n bodies, Ann. ofMath. 12 (1910), 1–17.

Planar central configurations

In the rest of this talk we are only interested

either in planar central configurations which are not collinear,

or in in spatial central configurations which are not planar.

For arbitrary masses m1, . . . ,mN the planar centralconfigurations are in general unknown when the number of thebodies N > 3. Numerically for N = 4 are known.

C. SIMÓ, Relative equilibrium solutions in the four–bodyproblem, Cel. Mech. 18 (1978), 165–184.

For arbitrary masses m1, . . . ,mN the spatial centralconfigurations are in general unknown when the number of thebodies N > 4.

Are there finitely many classes of planar centralconfigurations?

Wintner’s conjecture (1941): Is the number of classes of planarcentral configuration finite, in the N–body problem for anychoice of the masses m1, . . ., mN?

This conjecture also appears in the Smale’s list on themathematical problems for the XXI century.

M. HAMPTON AND R. MOECKEL, Finiteness of relative equilibriaof the four–body problem, Invent. Math. 163 (2006), 289–312.

A. ALBOUY AND V. KALOSHIN, Finiteness of centralconfigurations of five bodies in the plane, Ann. of Math. (2) 176(2012), 535–588.

The contens

Central configurations of the coorbital satellite problem

Now we want to consider a restricted version of the planarcentral configurations; i.e. we study the limit case of one largemass and n small equal masses when the small masses tendto zero.

This problem is called

either central configurations of the planar 1 + n–body problem,

or central configurations of the coorbital satellite problem.

The (1 + n)–body problem was first considered by

J.C. MAXWELL, On the Stability of Motion of Saturn’s Rings,Macmillan & Co., London, 1885.

In the (1 + n)–body problem the infinitesimal particles interactbetween them under the gravitational forces, but they do notperturb the largest mass.

The (1 + n)–body problem was first considered by

J.C. MAXWELL, On the Stability of Motion of Saturn’s Rings,Macmillan & Co., London, 1885.

In the (1 + n)–body problem the infinitesimal particles interactbetween them under the gravitational forces, but they do notperturb the largest mass.

Let r(ε) = (r0(ε), r1(ε), . . . , rn(ε)) be a planar centralconfiguration of the N–body problem with N = 1 + n,associated to the masses m0 = 1 and m1 = . . . = mn = ε,which depend continuously on ε when ε→ 0.

We say that r = (r0, r1, . . . , rn) with ri 6= rj if i 6= j and i , j ≥ 1, isa central configuration of the planar (1 + n)–body problem ifthere exists the limε→0 r(ε) and this limit is equal to r.

Let r(ε) = (r0(ε), r1(ε), . . . , rn(ε)) be a planar centralconfiguration of the N–body problem with N = 1 + n,associated to the masses m0 = 1 and m1 = . . . = mn = ε,which depend continuously on ε when ε→ 0.

We say that r = (r0, r1, . . . , rn) with ri 6= rj if i 6= j and i , j ≥ 1, isa central configuration of the planar (1 + n)–body problem ifthere exists the limε→0 r(ε) and this limit is equal to r.

From this definition it is clear that if we know the centralconfigurations of the (1 + n)–body problem, then we cancontinue them to sufficiently small positive values of ε.

PROPOSITION: All central configurations of the planar(1 + n)–body problem lie on a circle S1 centered at r0.

This is the reason for which the central configurations of the(1 + n)–body problem have applications to the dynamics of thecoorbital satellite systems.

PROPOSITION: Let r = (r0, r1, . . . , rn) be a centralconfiguration of the planar (1 + n)–body problem. Denoting byθi the angle defined by the position of ri on the circle S1, wehave

n∑j=1,j 6=k

sin(θj − θk )

[1− 1

8| sin3(θj − θk )/2|

for k = 1, . . . ,n.

A proof of both propositions can be found in

G.R. HALL, Central configurations in the planar 1 + n bodyproblem, preprint, 1988 (unpublished).

J. CASASAYAS, J. L. AND A. NUNES, Celestial Mechanics andDynamical Astronomy 60 (1994), 273–288.

n∑j=1,j 6=k

sin(θj − θk )

[1− 1

8| sin3(θj − θk )/2|

for k = 1, . . . ,n.

n∑j=1,j 6=k

sin(θj − θk )

[1− 1

8| sin3(θj − θk )/2|

for k = 1, . . . ,n.

Numerical results due to

H. SALO AND C.F. YODER, The dynamics of coorbital satellitesystems, Astron. Astrophys. 205 (1988), 309–327.

n Number of central configurations2 23 34 35 36 37 58 39 1

For the n’s of the table they also study the linear stability of thecentral configurations.

R. MOECKEL, Linear stability of relative equilibria with adominant mass, J. of Dynamics and Differential Equations 6(1994), 37–51.

Numerical computations seem to indicate that

(1) for n ≥ 9 the number of central configurations is 1 (we knownumerically that this is the case for 9 ≤ n ≤ 100).

(2) every central configuration has a straight line of symmetry.

J.M. CORS, J. L. AND M OLLÉ, Central configurations of theplanar coorbital satellite problem, Celestial Mechanics andDynamical Astronomy 89 (2004), 319–342.

Analytical results

(1) The numerical results of the table for n = 2,3,4, have beenproved analytically by

Euler and Lagrange for n = 2,Hall for n = 3, andCors, L. and Ollé for n = 4.Albouy and Fu (line of symmetry) for n = 4.

A. ALBOUY AND YANNING FU, Relative equilibria of fouridentical satellites, Proc. R. Soc. Lond. Ser. A Math. Phys.Eng. Sci. 465 (2009), 2633–2645.

Analytical results

(1) The numerical results of the table for n = 2,3,4, have beenproved analytically by

Euler and Lagrange for n = 2,Hall for n = 3, andCors, L. and Ollé for n = 4.Albouy and Fu (line of symmetry) for n = 4.

A. ALBOUY AND YANNING FU, Relative equilibria of fouridentical satellites, Proc. R. Soc. Lond. Ser. A Math. Phys.Eng. Sci. 465 (2009), 2633–2645.

(2) For n ≥ 2 it is known that the regular n–gon, having theinfinitesimal particles in its vertices and with the large mass inits center, is a central configuration.

(3) For n ≥ e27000 the regular n–gon is the unique centralconfiguration.

This was proved by Hall in 1988 in the mentioned unpublishedpaper.

This was proved in the quoted Casasayas, L. and Nunes paperof 1994.

CONJECTURES. For the central configurations of the(1 + n)–body problem:

(1) For n ≥ 9 the regular n–gon is the unique centralconfiguration.

(2) Prove that every central configuration has a straight line ofsymmetry.

We only need to prove conjecture (2) for 5 ≤ n ≤ 8, whenconjecture (1) be proved.

The contens

Double nested central configurations for the planar2n–body problem

I do not know who was the first in proving that the regularn–gon with equal masses is a central configuration.

W.R. LONGLEY, Some particular solutions in the problem ofn–bodies, Amer. Math. Soc. (1907), 324–335.

p = 2 and n = 2,3,4,5,6.

S. ZHANG AND Q. ZHOU, Periodic solutions for the 2n–bodyproblems, Proc. Am. Math. Soc. 131 (2002), 2161–2170.

p = 2 and n ≥ 2.

p = 2 and n = 2,3,4,5,6.

p = 2 and n ≥ 2.

p = 2 and n = 2,3,4,5,6.

p = 2 and n ≥ 2.

Triple and Quadruple nested central configurations forthe planar 3n– or 4n–body problem

J. L. AND L.F. MELLO, Triple and Quadruple nested centralconfigurations for the planar n–body problem, Physica D 238(2009) 563–571.

p = 3,4 and n ≥ 2,3,4.

Central configurations of p nested n–gons

THEOREM For all p > 2 and n > 2, we prove the existence ofcentral configurations of the pn–body problem where themasses are at the vertices of p nested regular n–gons with acommon center. In such configurations all the masses on thesame n–gon are equal, but masses on different n–gons couldbe different.

M. CORBERA, J. DELGADO AND J. L., On the existence ofcentral configurations of p nested n–gons, Qual. Theory Dyn.Syst. 8 (2009), 255–265.

Central configurations of p nested n–gons

THEOREM For all p > 2 and n > 2, we prove the existence ofcentral configurations of the pn–body problem where themasses are at the vertices of p nested regular n–gons with acommon center. In such configurations all the masses on thesame n–gon are equal, but masses on different n–gons couldbe different.

M. CORBERA, J. DELGADO AND J. L., On the existence ofcentral configurations of p nested n–gons, Qual. Theory Dyn.Syst. 8 (2009), 255–265.

The contens

Central configurations of nested regular polyhedra forthe spatial 2n–body problem

F. CEDÓ AND J. L., Symmetric central configurations of thespatial n–body problem, J. of Geometry and Physics 6 (1989)367–394.

THEOREM We consider 2n masses located at the vertices oftwo nested regular polyhedra with the same number of vertices.Assuming that the masses in each polyhedron are equal, weprove that for each ratio of the masses of the inner and theouter polyhedron there exists a unique ratio of the length of theedges of the inner and the outer polyhedron such that theconfiguration is central.

M. CORBERA AND J. L., Central configurations of nestedregular polyhedra for the spatial 2n–body problem, J. ofGeometry and Physics 58 (2008), 1241–1252.

Nested regular octahedra

Nested regular cube

Nested regular icosahedra

Nested regular dodecahedra

Central configurations of 3 nested regular polyhedrafor the spatial 3n-body problem

Idem with 3 nested regular polyhedra.

M. CORBERA AND J. L., Central configurations of three nestedregular polyhedra for the spatial 3n–body problem, J. ofGeometry and Physics 59 (2009), 321–339.

Nested regular tetrahedra

Nested regular octahedra

Nested regular cube

Nested regular icosahedra

Nested regular dodecahedra

On the existence of central configurations of p nestedregular polyhedra

THEOREM For all p > 2, we prove the existence of centralconfigurations of the pn–body problem where the masses arelocated at the vertices of p nested regular polyhedra having thesame number of vertices n and a common center. In suchconfigurations all the masses on the same polyhedron areequal, but masses on different polyhedra could be different.

M. CORBERA AND J. L., On the existence of centralconfigurations of p nested regular polyhedra, Celestial Mech.Dynam. Astronom. 106 (2010), 197U-207.

On the existence of central configurations of p nestedregular polyhedra

THEOREM For all p > 2, we prove the existence of centralconfigurations of the pn–body problem where the masses arelocated at the vertices of p nested regular polyhedra having thesame number of vertices n and a common center. In suchconfigurations all the masses on the same polyhedron areequal, but masses on different polyhedra could be different.

M. CORBERA AND J. L., On the existence of centralconfigurations of p nested regular polyhedra, Celestial Mech.Dynam. Astronom. 106 (2010), 197U-207.

Central configurations of nested rotated regulartetrahedra

Idem but rotated with 3 nested regular polyhedra.

M. CORBERA AND J. L., Central configurations of nestedrotated regular tetrahedra, J. of Geometry and Physics 59(2009), 137–1394.

1 6 ρ < R

1 < R 6 ρ

0 < ρ < 1 < R

The contens

From the paperW.D. MACMILLAN AND W. BARTKY, Permanent Configurationsin the Problem of Four Bodies, Trans. Amer. Math. Soc. 34(1932), 838–875.we have the following two conjectures:

CONJECTURE 1 The 4–body problem has a unique class ofconvex central configuration.

CONJECTURE 2 If the 4–body problem has a centralconfigurations with two pairs of equal masses located at twoadjacent vertices of a convex quadrilateral, then thisquadrilateral is an isosceles trapezoid.

Conjecture 2 has been proved recently in the paper:

A.C. Fernandes, J. L. and L.F. Mello, Convex centralconfigurations of the 4–body problem with two pairs of equalmasses, Archive for Rational Mechanics and Analysis 226(2017), 303–320.

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